270 XIV INTEGRATION IN LOCALLY COMPACT GROUPS each s<=G, there exists an integer n such that l^n^m and (If 77 : G -»• G/H is the canonical mapping, observe that /x( 7r(U)) > 0 and that the measures of the sets sj • 7r(U) are all equal.) 5. Let G be a locally compact group and ^ a right Haar measure on G. (a) In order that a positive measure a ^ 0 on G, with support S, should be such that S is a closed subgroup of G and the measure induced on S by a a right Haar measure on S, it is necessary and sufficient that 5Cs)a = a for all s e S. The set of elements / e G such that 5(r)a = a is then equal to S. (b) Let F denote the set of positive measures a on G satisfying the condition of (a) above. Show that T is vaguely closed in M+(G)- {0}. If /e JT+(G) is such that f(e) > 0, show that the set F/ of measures a e T such that m» doc(s) — I is vaguely compact (prove that sup a(K) < + oo for every compact subset K of G). Deduce that a e r/ the mapping ai—>(a(/), a/a(/)) is a homeomorphism of T onto the product space R*+ xiy (c) For each a e F, let H« = Supp(a) : it is a closed subgroup of G. Let d be a right- invariant distance on G such that d(x, y)^l for all x, y in G (12.9.1). Let (Kn) be an increasing sequence of compact sets, covering G and such that Kn is contained in the interior of Kra+1 for each n (3.18.3). If h is the HausdorfT distance on G corresponding to d (Section 3.16, Problem 3), then for any two nonempty closed sets M, N in G we define hn(M, N) = 1 if either M r\ Kn or N n Kn is empty, and /rn(M, N) « h(M r\ Kn, N n Kn) otherwise. Endow the set fHG) of nonempty closed subsets of G with the topology defined by the pseudo-distances hn. Show that the mapping a h-> H« is a homeomorphism of F/ (endowed with the vague topology) onto the set 2 of closed subgroups of G, endowed with the topology induced by that of 6. With the notation of Problem 5, consider the subset F° of F consisting of measures a e F such that Ha is unimodular. This is also the set of measures a e F such that a(/) = a(/) for all /e Jf(G), and is therefore vaguely closed in F. For each a e F°, put Qa = G/H«; then there exists a relatively G-invariant measure ^ on Q« such that f f(x)d^(x)= f d^(x) ( f(xs)doc(s) JG JQa ^Ho for every /e ^T(G) (where x is the coset xUa) (Problem 2). (a) If a 6 F° and /e JT(G), put /«(*) = f f(xs) da(s). Show that the mapping JH<X ai— > ll/all is vaguely continuous, and deduce that the mapping ai— > ||jiia|| is lower semi- continuous with respect to the vague topology. (b) Let g ^ 0 be a real-valued jit-integrable function, and let F°(#) be the set of measures a e F° such that I g(xs) du(s) g: 1 for all x e G. Show that the mapping ai—>> ll/i-all of F°(^) into R is vaguely continuous, (It is enough to show that the map- ping is upper semicontinuous. Let h e JT+(G) be such that I \g(x) — h(x)\ dp.(x) ^ e, and let K = Supp(/z). If TT^ : G -> Q^ is the canonical mapping, show that fiff(Q0 - 77>(K)) S £ for all f$ E T°(g). On the other hand, consider a f unction /eJf+(G) such that f f(xs) da(s) <i 1 for all s e G and f f(xs) da(s) = 1 for all s e K (use Prob- JG JG lem 2(a)). If Ue is the set of measures j8 e F°(^) such that f f(xsdft(s)> 1 - e for all x eK, show that (1 - e)^(ir/K)) < ||f*.||.) 6).pology «^~4, v (use (e)). Give an example of a